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NAG Toolbox: nag_roots_withdraw_contfn_brent_int (c05ad)

Purpose

nag_roots_withdraw_contfn_brent_int (c05ad) locates a zero of a continuous function in a given interval by a combination of the methods of nonlinear interpolation, linear extrapolation and bisection.
Note: this function is scheduled to be withdrawn, please see c05ad in Advice on Replacement Calls for Withdrawn/Superseded Routines..

Syntax

[x, ifail] = c05ad(a, b, eps, eta, f)
[x, ifail] = nag_roots_withdraw_contfn_brent_int(a, b, eps, eta, f)

Description

nag_roots_withdraw_contfn_brent_int (c05ad) attempts to obtain an approximation to a simple zero of the function f(x) f(x)  given an initial interval [a,b] [a,b]  such that f(a) × f(b) 0 f(a) × f(b) 0 . The same core algorithm is used by nag_roots_contfn_brent_rcomm (c05az) whose specification should be consulted for details of the method used.
The approximation xx to the zero αα is determined so that at least one of the following criteria is satisfied:
(i) |xα| eps |x-α| eps ,
(ii) |f(x)|eta |f(x)|eta .

References

Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall

Parameters

Compulsory Input Parameters

1:     a – double scalar
aa, the lower bound of the interval.
2:     b – double scalar
bb, the upper bound of the interval.
Constraint: ba ba .
3:     eps – double scalar
The termination tolerance on xx (see Section [Description]).
Constraint: eps > 0.0 eps>0.0 .
4:     eta – double scalar
A value such that if |f(x)|eta |f(x)|eta , xx is accepted as the zero. eta may be specified as 0.00.0 (see Section [Accuracy]).
5:     f – function handle or string containing name of m-file
f must evaluate the function ff whose zero is to be determined.
[result] = f(xx)

Input Parameters

1:     xx – double scalar
The point at which the function must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     x – double scalar
The approximation to the zero.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail = 1ifail=1
On entry, eps0.0 eps0.0 ,
or a = b a=b ,
or f(a) × f(b) > 0.0 f(a) × f(b) > 0.0 .
W ifail = 2ifail=2
Too much accuracy has been requested in the computation; that is, the zero has been located to relative accuracy at least εε, where εε is the machine precision, but the exit conditions described in Section [Description] are not satisfied. It is unsafe for nag_roots_withdraw_contfn_brent_int (c05ad) to continue beyond this point, but the final value of x returned is an accurate approximation to the zero.
W ifail = 3ifail=3
A change in sign of f(x) f(x)  has been determined as occurring near the point defined by the final value of x. However, there is some evidence that this sign-change corresponds to a pole of f(x) f(x) .

Accuracy

The levels of accuracy depend on the values of eps and eta. If full machine accuracy is required, they may be set very small, resulting in an exit with ifail = 2ifail=2, although this may involve many more iterations than a lesser accuracy. You are recommended to set eta = 0.0 eta=0.0  and to use eps to control the accuracy, unless you have considerable knowledge of the size of f(x) f(x)  for values of xx near the zero.

Further Comments

The time taken by nag_roots_withdraw_contfn_brent_int (c05ad) depends primarily on the time spent evaluating f (see Section [Parameters]).
If it is important to determine an interval of relative length less than 2 × eps2×eps containing the zero, or if f is expensive to evaluate and the number of calls to f is to be restricted, then use of nag_roots_contfn_brent_rcomm (c05az) is recommended. Use of nag_roots_contfn_brent_rcomm (c05az) is also recommended when the structure of the problem to be solved does not permit a simple f to be written: the reverse communication facilities of nag_roots_contfn_brent_rcomm (c05az) are more flexible than the direct communication of f required by nag_roots_withdraw_contfn_brent_int (c05ad).

Example

function nag_roots_withdraw_contfn_brent_int_example
a = 0;
b = 1;
epsilon = 1e-05;
eta = 0;
f = @(x) exp(-x)-x;
[x, ifail] = nag_roots_withdraw_contfn_brent_int(a, b, epsilon, eta, f)
 

x =

    0.5671


ifail =

                    0


function c05ad_example
a = 0;
b = 1;
epsilon = 1e-05;
eta = 0;
f = @(x) exp(-x)-x;
[x, ifail] = c05ad(a, b, epsilon, eta, f)
 

x =

    0.5671


ifail =

                    0



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Chapter Contents
Chapter Introduction
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